Getting More from MooreG. Edgar Parker

G. Edgar Parker wrote a Ph.D. thesis in analysis under the direction of John W. Neuberger, a student of H.S. Wall. He is Professor of Mathematics at James Madison University.

ABSTRACT: From the basis of pedagogy pioneered by R.L. Moore (the Moore Method), Moore-style pedagogy is discussed, with special attention being paid to techniques likely to make the method Moore inclusive. Ramifications of extending the mindset behind the Moore method to service courses are also discussed.

KEYWORDS: Moore method, discovery learning.

This paper is not offered as a work of science; indeed, whether or not pedagogical considerations are best treated with scientific methods is an arguable point. Neither is the work offered whimsically; the experiences I have had as a student at the undergraduate and graduate levels and as a secondary teacher, as a graduate assistant, and as a university professor in both open admissions and selective institutions have exposed me to many different teaching styles and given me both cause and context to experiment extensively with non-lecture methods. it is my hope that sharing the strategies that I have adopted or rejected might give food for thought to fellow teachers or ideas for investigation to those who might view pedagogy as science. Many of the ideas are likely already well-traveled; I make no claims of priority even though most of what I will discuss has become a part of my own tactics through trial and error with my own teaching.

The Moore method as R.L. Moore practiced it and simplistically interpreted, might be characterized as the practice of

"causing students to perform research at their own level (emphasis mine) by confronting the class with impartially posted questions and conjectures which are at the limits of their capability." [8]

In [8], Traylor lists further characteristics of Moore's method and takes great care to note the personal judgments Moore was willing to make such as segregating his classes relative to mathematical experience and "sacrificing" a class for the benefit of a particularly talented individual. Traylor warns that technique may only "suggest the method," since Moore's successes may have been so dependent upon his own force of character. Traylor even warns of those who might "weaken the application of the method and reduce the dividend" or attempt the technique without proper commitment and produce "practices far removed from Moore method (but) go by that name."

With this admonition in mind, let me introduce myself so that you can have a background for what type of charlatan I may be and describe for you the way I will use the phrase Moore method. I was in a Moore-method mathematics classroom every semester and quarter I spent in higher education. My only exposure to lecture-method mathematics teaching is from high school, the calculus sequence and differential equations as an undergraduate, the algebraic structures sequence, linear algebra, and model theory as a graduate student, and the time I have spent discussing and observing teaching with my colleagues. I was extremely disillusioned with Moore-method teaching after taking the GRE's and not being able to read most of the questions; I was reconverted as a graduate student when I experienced the lecture phenomenon anew and was able to compare the nature of my own preparation with that of my fellow students. I have used the Moore-style classroom in my upper level and graduate courses throughout my university teaching career. I have also tried to incorporate Moore-style strategies into my "lecture" courses. I make no claims of universal success--sometimes courses with which I was pleased left the students unhappy; some courses with which the students were pleased left me dissatisfied. At least two courses I have given using the Moore method were unmitigated failures by any standard whatsoever. Nevertheless, the successes have occurred often enough and dramatically enough to reinforce my belief that non-lecture style teaching is best suited to making mathematics and effective and vital part of the students' intellectual repertoires.

A student is very much the product of his or her mentors; let me pay tribute to those teachers with whom I took Moore-method courses. These are the persons who served as models and from whom I have both borrowed strategies and personas and identified qualities I hoped to avoid as a teacher. To all of them I am indebted. They are J. R. Boyd, Ken Walker, and Elwood Parker of Guilford College; and David Ford, William Mahavier, John Neuberger, Phil Tonne, and Mary F. Neff of Emory University. Special thanks go to J. R. Boyd, who first exposed me to the method; William Mahavier, whom I consider the finest practitioner I have observed of the method; and John Neuberger, my thesis advisor, who brought human qualities to teaching that remain the ideal for which I strive.

I was not taught by Moore, but I have been taught by three of his students. My perceptions of him have come through anecdotes and from [2] and [8]. In this paper, Moore method should not be construed as Moore's method; indeed; I wish neither to pay tribute to nor disavow the work of Moore. I use the term "Moore method" as a name, a name attached to the pioneer of the practice of a mind set from which a style of teaching has emerged. The Moore method, as I use the term, is not necessarily Moore's method; rather it is a commitment to teaching by letting students discover the power their own minds have. As a colleague, Bill Sanders, once put it--"Our job as teachers is to get the students to realize that they don't need teachers."

Granted Bill's articulation as a worthy goal, how then should we go about "our job"? Let us first address the problems of teaching students who have an expressed interest in mathematics. This is the context in which the Moore method has been granted its greatest acceptance. There are at least three stages through which students specializing in mathematics are typically led:

(i) making proofs,

(ii) mastering a core of pertinent material, and

(iii) venturing out to pose questions not generated by the courses themselves.

Item (i) is typically addressed during the undergraduate experience, item (ii) is the core of undergraduate and graduate experience, and item (iii) is basically (though not exclusively) a graduate phenomenon.

Even critics of Moore-style strategies generally grant the efficacy of addressing item i. with the Moore method. They often argue, however, that it is done with significant sacrifice to the coverage of material and that only the best students can learn this way. Under the premise that there might be some credibility to these contentions, let us address them.

It is quite possible that the trade-off in coverage of material for added experience through the student's added use of her or his own devices is a good one [1]; nevertheless, it is also reasonable to try to maximize the amount of material covered. Maximizing coverage may come from at least two sources--

creating a context in which a lot of problems get solved, and

having problems that, once solved, make other problems accessible.

Even in well-written textbooks, all sections are not created equal. Making value judgments as to relative worth is a first step in creating a course to be presented by the Moore method. Given a body of material that, if mastered, might constitute a course (a syllabus?), the initial question I ask is "which theorems would guarantee a bare minimum for a course that I would be willing to put my name on?" Once this question is answered, the course is made around the theorems selected, the "main theorems." The "bare minimum" provides a skeleton for the course, the other problems are then to be used to accomplish proofs for the skeletal problems or to continue the course from the point where the minimal goals are achieved.

The first priority after selecting the main theorems is to blaze a trail to them. When doing a course for the first time, chocies can be made by answering the question "what did it take for me to understand this theorem?" The answers should allow you to generate problems which are more accessible than the main theorem, thus creating an opportunity for more "successes." If the lead-in problems could, if solved the way you envision them being solved, also isolate different proof techniques, so much the better. Be warned, however, that if your experience is the same as mine, the one constant of using Moore method is that students will as often as not develop techniques different than what you had in mind to solve your lemmas and even build their own roads to the main theorems. It is for this reason that I never distinguish between lemmas and the theorems the lemmas suggest. To a mathematics student experiencing her or his first success, I see no justification for cheapening the thrill by deeming some problems as preliminaries. In building morale, it is far better to be able to look back and exult about how "________ was able to use your and ________'s theorems to help prove this one!" Also, if preliminary problems are designated as such, the chances of a student creating a proof other than what you had in mind are diminished.

Observing the way that students solve your problems holds one key to improving the course notes in subsequent offerings, since it offers the option of choosing the main ideas in the student development as the preliminary problems instead of what you originally had. As delightful as it is to see students uncover things that you had not planned, be mindful that certain proof techniques may be important components of the content of the course. If students do not discover them where you had intended, new problems need to be stated that provide second opportunities to uncover the techniques. Although courses are planned in their entirety in advance, it is important that the actual notes evolve with the course so that the strengths of the students can be followed and the struggling student not be abandoned.

This brings us to another common criticism of the Moore method--that only the best students can benefit from it. The commitment to being a pump, not a filter" [4] is an important one. Perhaps if the approach is to state the geometry axioms for the numbers and state as Problem 1 the Heine-Borel Theorem for an interval, this criticism would be justified. But in the program outlined above the layering of problems allows you to state your problems at any or all difficulty levels you choose. The prime ingredient is to produce a context for success. My experience indicates that just the rudiments of logic are painful for even good students until they have been forced to use them first hand. Whatever it is about the logic of quantification that is difficult really is difficult until a student actually uses it. Blatantly obvious (at least from the point of view of the instructor) theorems can make good problems if the student has to produce an argument that evokes the use of the logic of quantification in order to make the proof. Thus I try to include problems where the most difficult aspect is the recognition that an object must be exhibited (for existentials) or that the argument must be made for "each such object" rather than "this particular object" (for universals).

We must also remain aware that sophistication grows at different rates within classes. As each new topic is revealed, I try to include problems that I know can be done with one- or two-step proof so that the fledgling will always have accessible problems available. In the other direction, it is also important that the students who are more mathematically mature coming in or who mature more rapidly during the course have access to problems which leave bigger gaps between hypothesis and conclusion. The teacher can direct students to problems she or he deems appropriate without pushing or insisting by using conversational devices such as "have you tried Problem ____ yet?" or "from what I saw you do on Problem _____, I'll bet Problem ____ will interest you".

One of the greatest strengths of the Moore method, one of which I was not even aware when I first started using the method, is the arena it provides for students to do the type of learning that they do in a traditional lecture format. As a student I was always obsessed with being the first to solve problems; I considered having to listen to someone else's argument evidence of failure on my part (although I never had the courage not to listen!). As a teacher I recognize that the next best thing to solving a problem is to have worked on it, and then to see what it took to get past the point on which one was stuck. A close third behind this is to closely question an argument, then to reproduce the thinking without the benefit of notes.

I try to encourage these behaviors through the way credit is offered. I use a "call-on" class structure. The roll is randomly ordered, then each student, in turn according to the order, is offered the opportunity to present a problem of her or his choice. The student may refuse her or his turn with no penalty other than having to wait through an entire cycle of the roll, or may make an oral presentation. If the class judges the work correct (I am a member of the class!), the student is awarded double credit. If the student has an error that she or he cannot correct, no penalty is exacted. I will typically try to highlight where the difficulty has occurred and try to praise something accomplished before the snafu or praise the method of attack. This is my response to Jones's [2] warnings about the trauma associated with being wrong in public; to prevent criticism from being taken as ridicule it can be reassuring to have to last thing heard be praise. The problem remains available to anyone else who might solve it. Particularly positive things tend to happen when someone settles a problem that others have tried and the teacher is able to point out how earlier work seems to have been incorporated into the argument. This tends not only to defuse the despair of those who tried the problem earlier, but also to reinforce the notion that mathematics may be built bit-by-bit as well as in flashes of insight. To a student being forced to encounter the need to create rather than to just recreate, this can be a very reassuring experience.

The strongest secondary effect of Moore-method teaching is that students make good mistakes; that is, the mistakes are good in the sense that they are not contrived. Students think like students think, not like a professor trying to think like a student but who can't because the naivete of beginning mathematics has been overwhelmed by education and experience. Indeed, the mistakes often do the teaching. Not only is there usually some other student with the same misconception, but mistakes provide a great source for problems which are feasible in the mind of the student, but which are not theorems. A spirited case for the inclusion of such problems, although not necessarily student-generated, is offered by McNerney [3] in the preface to his book on complex analysis. In addition, even when a student offers a "correct" argument, the presentation is seldom polished. This leaves the rest of the class something to do. I hold my class responsible for anything that goes on the board and is deemed correct. I guarantee that there will be enough questions on the examination straight from the course for the student to pass (with a C if the course is content-based rather than a learn-to-make-proofs course) and that the students may use their notes on the examination. In order to discourage the students from taking notes so that they will have to rethink as well as smooth out the problem when they prepare their own written versions, I promise to reproduce in my office anything that occurs on class as many times as they want to see it. To reward whatever successes students have whenever they have them, I offer full credit for a correct written solution so long as it was written before the problem was presented and turned in the day of the presentation. This is a particularly nice incentive when problems are interrupted by the end of a class period since students who might not have had an idea are given a beginning on a problem. Having students write and receive feedback is also important pedagogy. In oral presentations students can improvise to patch mistakes; written presentations offer no such second chances. Even if a paper is correct, the opportunity is present to remark in ways that can help a student write with greater clarity.

Unless a teacher works in a department where most of the courses are given by the Moore method (in which case she or he should seriously consider switching to lecture), the teacher should be aware that there is likely to be considerable student reaction to the differences between the Moore-method classroom and what the student has likely experienced before. In my own classes I have had student evaluations that ran from "why aren't all courses offered this way?" to "Dr. Paker has violated the contract between the university and the student...". Although I make no apologies for the method to anyone, let alone the students, I do try to alert the students at the beginning of the course to the fact that discovering why a theorem is correct may be a much more challenging experience than understanding why an argument for a theorem is correct. When a student's fears become reality and the student begins to struggle, it is important to play a role as a morale builder. As mentioned before, problem sets are designed to accommodate differing success timetables and I try to reduce loss of face by finding something correct in a lost argument. In addition, triumphs need to be reinforced. I try to do this by finding parts of arguments that stand alone and naming the resulting theorems after the founders or by making such a recognition when a student reduces a problem to another problem, whether the student solves the problem or not. I try to devote the greatest part of my classroom energy to being a cheerleader rather than a critic. (I take it as a sign that I am being too active a critic when the students look at me to gauge my reaction before addressing the presenter, and consider the day-to-day decisions surrounding how active a critic to be the most difficult aspect of Moore-method teaching.) However, be assured that weight of personality will have little effect unless the students experience success. Whyburn's advice is as pertinent today as when Jones [2]
offered it--"give them something they can do."

This admonition returns us to the problem of structuring course notes. My experience is that if a course can be based around a model for the axioms being studied without sacrificing its integrity, then it should. In my course on the numbers, for instance, rather than proceeding from the axioms, I choose to make either the binary or fraction model for the numbers and try to get the students to prove the axioms from the model and explore the ideas that are consequent to the axioms from the model. One definite advantage of this approach is the ease with which existential problems can be concocted, a boon since learning to make constructions to solve existential problems appears to me to be the most difficult task beginning mathematics students face. Even in courses where the axiomatic structure justifiably occupies center state, I try to have at least some second strand of problems associated with some embodiment of the axioms. For example, in analysis, the study of continuous number functions can be accompanied by the study of some particular functions rather than simply using them as applications for the theory when the theory is complete. In geometry, problems showing that some model or models make the axioms true is a nice complement to producing the consequences of the axioms.

One last debt is owed to coverage. At the conclusion of any course, I try to make whatever translations might be necessary for the student to get into the literature. I will also recommend books that cover the same material that we have covered. I offer myself as a source if the student wants to work further or wants to try to make connections with subsequent courses.

Courses with broader audiences present different challenges to using the Moore method. Beginning courses are often stocked with students whose only interest appears to be to get a required mathematics credit and whose primary commitment seems to be to pull a passing grade with as little work as possible. Service courses are often dictated by syllabi that seem to be largely geared to putting a technique in a student's hand in time for application in the course being served. Nevertheless, the principles that guide the structure of Moore-method courses are still applicable. I have found that trying to do something about what I perceive to be flaws in the textbooks or the tyranny of the syllabus can bring Moore-style principles into play.

The principle of layering, of identifying priorities among the topics as discussed earlier, is critical if time is to be bought for use as discovery time. Although we should honor our departments' commitments to coverage, we can still identify priorities and make the "big deal" problems the focal points while using the side issues to reinforce them. There is nothing sacred about textbook organization. I try to organize a course so that the priority problems appear as often as possible. Instead of "motivating idea--development of the theorems--applications of the theorems" forming the structure of the sections or chapters, let it form the structure of the course.

There are several advantages to making this the gross structure rather than the fine structure of a course. First of all, a context for separating structure from application is delineated. The mathematics that is done in making careful articulation of the ideas gleaned from the motivating problem can be separated in space and time from the uses of the theory that follows. This allows for emphasis on the use of language as a part of mathematics in contrast to the discussive mode often used in thinking about mathematics. It may well also increase the opportunities to spiral material ([5], [6], and [7]). Giving the students the wherewithal to talk within a formal setting also makes the making of proof a possibility.

Most textbooks are loaded with behavioristic learning traps. Advertisements boast of clearly presented sample problems and a text's plethora of solved exercises. Is the place of mathematics in the liberal arts curriculum its emphasis on procedure? Is proper preparation for the use of mathematics emphasis on pattern-matching and imitation? I think not, and if I am correct, then mathematics instruction needs to be based on the students being able to justify their own conclusions. This need not be done through a theorem-proof format. Exercises such as are typically found at the end of sections in books are an excellent context, but the questions about them must be asked directly and completely so that the student knows what is expected. The goal needs to be changed from "get the answer in the back of the book" to "reach a conclusion and justify it." Justification may be the instantiation and application of a theorem, or it may require reasoning from a principle established within the mathematics. The important thing from a Moore-method viewpoint is that the student be left with a question, not a prototype. Pace can be maintained by following up on the questions. If good incentives are found, answers can be put in place by student response or teacher response to student questions. This is to be preferred to a teacher anticipating what student questions might be.

Of course, textbooks are not typically written with this in mind--rehearsal and repetition still dominate textbook organization. One way to defeat the answers int he back of the book is to use the textbook exercises but to give different directions. The resulting problems are practically always interesting since they are not rigged to be computationally convenient. In addition, questions are likely to come from students who have answers but are not sure they are correct. This affords an opportunity to give a context for the reasonableness of making proofs that is student, rather than faculty, generated. "Then why do you think this is an answer?" has been the lead questions in many fruitful classes.

Circumventing the fact that proofs are given to the main theorems is not quite so simple. For some reason, textbook writers seem intent on proving the accessible theorems, then leaving subtlety and clever twists for the starred exercises and the students. This seems backwards to me. I try to assign problems, that, if solved are likely to come from thinking that will translate into proofs for the theorems. In discovery teaching, it is more important to recognize that what certifies the correctness of a problem is likely to be the key to articulating a principle from which the proof of a theorem can be made. This is a different emphasis than become adept at applying the theorems to whatever problems that obey their hypotheses. This is not to say that the latter is not important, but if the theory is to be more than an extra long list of axioms, the former must play an important part in the development of the course. Sometimes skipping around for exercises can be used to distract students from the text long enough to ensure unbiased attempts at the key problems. However, just as in non-textbook courses, you owe your students a chance for literacy. At the proper time (hopefully after they have made the discovery themselves), they should be pointed to the textbook treatment of their work. The difference is that the book is the third source, to follow individual investigation and teacher follow-up.

Much has been said (for example [4]) about guaranteeing a literate mathematical population for this country in the coming decades. A neglected audience is those students who have made up their minds that their future use of mathematics will be balancing checkbooks and who equate mathematics with such exercises. These are the people who are willing to proudly testify to anyone who will listen that they hate mathematics. Assuming that such people will indeed talk and probably reproduce as well, it is important that, even if we do not change such attitudes, we give the students a good look at the nature of mathematics as a viable mode of thinking and as the producer of many useful thoughts. At least then they can hate mathematics for the right reasons. Good mathematics can be done at any level of sophistication. I often use number algebra, with which all of our students have come into contact, as a context for discovery of the difference between procedure and verification in our "math for students who don't want to take math" course. The great advantage of teaching material to which there has been previous exposure over new material is that students have preconceived notions to examine and thus the students can be given a stake in the material about which they are asked to think. They are often eager to show you "how to do it," which can provide a good place to sample real understanding or lack of understanding. Thus discovery techniques can be used to create an appreciation for the discipline as well as to attract students to it.

The function of some courses as service courses can be used to complement the goal of getting students to do their own mathematics. Although at least one compelling application is important to motivating the articulation of the basis of each topic, I like to segregate the rest of the applications (see [6] and [7] for concrete applications of the plan) from the mathematics that supports them and dump them into a unit together. This supports pedagogy in at least two significant ways. Since the problems are isolated from the mathematics that solves them, the use of a model to pose the mathematical question becomes an important tool in determining which mathematics to use. The students are forced to make distinctions through argument since they can draw no clues from the sections in which the problems are found. Even the applications from college algebra require careful articulation when removed from the context of the type of equation that will solve them. The second nice aspect of this type of grouping is that, with all the mathematics in place, you know exactly how much time you have left to spend and the students can be turned loose on the problems at the point in the course when they should be most mature. Coverage is no longer an issue since the substance of the applications will be taught in the courses where they appear and the mathematics necessary for their solution is already in place. In fact, I pick my applications according to whether or not the principle necessary to make the application is contained in the problem. "Applications" such as "The function f describes the relationship between. Find the value at which cost is greatest." are bogus since the model is given. These problems have already been done when you simply state a function and optimize it. For an application to encourage discovery learning, it must require a translation from the context of the statement into some part of mathematics.

In conclusion, one final aspect of Moore-method teaching should not be discounted--the students' actual reliance on the teacher is limited to the teacher being a source of problems. Confidence grows exponentially with successes that can be attributed not to the clarity of some professor's exposition, but to the student's own guile. If "our job" is indeed to teach them they don't need teachers, then the Moore method remains a viable option.

Ed Parker is Associate Professor of Mathematics at James Madison University. He received his AB from Guilford College in 1969, and a PhD at Emory University in 1977 under John Neuberger. Although his interests are eclectic and his attention frequently dominated by baseball and other sports, his published research has centered around the study of the structure of the semigroups associated with the solutions of nonlinear differential equations. His teaching experience includes teaching mathematics at a public secondary school, junior college, an open admission university, and selective universities.